Is there a straightforward way to map all my vertices from a cubic space to a cylindrical space?
Squircle and shapekey

Subdivided cube run thru script to give squircle generated cylinder shapekey.
In How can I morph a flat plane to be a flat cirlce?  I have shown one of a number of square to circle mappings.
The mapping maps XY coords to a radius 1 circle, result shown above using default cube.  For radius not equal to one would have to scale and rescale accordingly, or make radius an argument of mapping.
Note: This is the squircle mapping, as demonstrated I did go thru and provide an example of all the mappings, if I find the file will add the conformal mapping. As explained in link in linked answer.
import bpy
import bmesh
from math import sqrt
from bpy import context
collection = context.collection
ob = context.object
me = ob.data
def squircle(x, y):
    u = x * sqrt(1 - y * y / 2)
    v = y * sqrt(1 - x * x / 2)
    return u, v


sk = ob.shape_key_add(name=&quot;Basis&quot;)
ci = ob.shape_key_add(name=&quot;Cylinder&quot;)    
for v in me.vertices:
    ci.data[v.index].co.xy = squircle(*v.co.xy)

Note: Changing space would be more akin to changing Cartesian x, y, z to cylindrical r, theta, h where r is the radius, theta the angle around the axis and h the height of the cylinder.  This wouldn't change the look.
Casting to Cylinder via modifier
Could instead add a cast modifier

Result on subdivided default cube, radius set to sqrt(2)
Not sure the top is the result desired, to which would possibly require using a weighted vertex group for interior verts with cylinder axis aligned normals.